92, 93] THREE TYPES OF PATH. 283 



case in Figs. 95, 96, 97, with the horizontal projections marked a, 

 while the cusped type for the second case is shown in Fig. 98. 



Fig. 98. Pig. 98 a. 



93. Precession and Nutation. It will be convenient to express 

 the motion in terms of one root 8 and the constants &, /5. In order 

 to eliminate cc we have 



98) - 

 Now since 8 l is a root, f(z) = and 



99) o a*- 

 Subtracting from 98), 



10(V> f(z} - A i (/?-K) 2 (i-* 2 ) 



TT?i**(A^ :jr H 1 _^ 2 



We thus find that ^ is a factor of the expression on the right, 

 so that, multiplying by 1 # 2 , we have f(z) exhibited in the form 



where {^(z) is the polynomial of order two, 



so that the other two roots are found by solving the quadratic 

 (0) = 0. As the roots z l9 s^ approach each other, the rise and fall 

 decreases, and vanishes when f(z) has two equal roots. The condition 

 for this is that /(*) and f (0) = fa (0) + (0 *i)fi(*) have a common 



root a., that is that 



from which 



103) a(l-^) 2 



If % and one of the constants 6, /3 are given, this is a quadratic to 

 determine the other. We find 



